The pH value of `10^(-7)` M solution HCI is

Options

A. Equal to 1
B. Equal to 2
C. Less than 7
D. Equal to 0

Answer

To find the pH value of a \(10^{-7}\) M solution of HCl, we need to consider the dissociation of HCl in water and the contribution of H+ ions from both HCl and water.

### Step-by-Step Solution:

1. **Understanding the Dissociation of HCl**:
HCl is a strong acid and completely dissociates in water:
\[ \text{HCl} \rightarrow \text{H}^+ + \text{Cl}^- \]
Therefore, the concentration of \(\text{H}^+\) ions from HCl is \(10^{-7}\) M.

2. **Considering Water's Contribution**:
Water also dissociates slightly into \(\text{H}^+\) and \(\text{OH}^-\):
\[ \text{H}_2\text{O} \rightleftharpoons \text{H}^+ + \text{OH}^- \]
The ion product of water (\(K_w\)) at 25°C is:
\[ K_w = [\text{H}^+][\text{OH}^-] = 10^{-14} \]
Let the concentration of \(\text{H}^+\) from water be \(x\). Therefore, the total concentration of \(\text{H}^+\) ions in the solution will be:
\[ [\text{H}^+] = 10^{-7} + x \]
And since \([\text{OH}^-] = x\), we can write:
\[ (10^{-7} + x)x = 10^{-14} \]

3. **Setting Up the Equation**:
Expanding the equation gives:
\[ 10^{-7}x + x^2 = 10^{-14} \]
Rearranging it, we have:
\[ x^2 + 10^{-7}x - 10^{-14} = 0 \]

4. **Using the Quadratic Formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 10^{-7}\), and \(c = -10^{-14}\). Plugging in these values:
\[ x = \frac{-10^{-7} \pm \sqrt{(10^{-7})^2 - 4 \cdot 1 \cdot (-10^{-14})}}{2 \cdot 1} \]
\[ = \frac{-10^{-7} \pm \sqrt{10^{-14} + 4 \cdot 10^{-14}}}{2} \]
\[ = \frac{-10^{-7} \pm \sqrt{5 \cdot 10^{-14}}}{2} \]
\[ = \frac{-10^{-7} \pm \sqrt{5} \cdot 10^{-7}}{2} \]

5. **Calculating the Positive Root**:
We take the positive root since concentration cannot be negative:
\[ x = \frac{-10^{-7} + \sqrt{5} \cdot 10^{-7}}{2} \]
Approximating \(\sqrt{5} \approx 2.236\):
\[ x \approx \frac{-10^{-7} + 2.236 \cdot 10^{-7}}{2} \approx \frac{1.236 \cdot 10^{-7}}{2} \approx 0.618 \cdot 10^{-7} \]

6. **Calculating Total \([\text{H}^+]\)**:
Now we can find the total concentration of \(\text{H}^+\):
\[ [\text{H}^+] = 10^{-7} + 0.618 \cdot 10^{-7} \approx 1.618 \cdot 10^{-7} \]

7. **Calculating pH**:
Finally, we calculate the pH:
\[ \text{pH} = -\log(1.618 \cdot 10^{-7}) = -\log(1.618) - \log(10^{-7}) \]
\[ \approx -0.209 + 7 \approx 6.791 \]

### Conclusion:
The pH of a \(10^{-7}\) M solution of HCl is approximately **6.79**, which is **less than 7**.